Solve-the-system-x-C-y-1-20-x-1-C-y-10-

Question Number 57074 by Tawa1 last updated on 30/Mar/19

Solve the system .

\overset{x}{} C_{y + 1} = 20, \overset{x - 1}{} C_{y} = 10

Answered by mr W last updated on 30/Mar/19

\overset{x - 1}{} C_{y} = 10

\Rightarrow \frac{(x - 1)!}{(x - 1 - y)! y!} = 10

\overset{x}{} C_{y + 1} = 20

\Rightarrow \frac{x!}{(x - y - 1)! (y + 1)!} = 20

\Rightarrow \frac{x (x - 1)!}{(x - 1 - y)! (y + 1) y!} = 20

\Rightarrow \frac{x}{(y + 1)} \times 10 = 20

\Rightarrow \frac{x}{y + 1} = 2

\Rightarrow x = 2 y + 2

\Rightarrow \frac{(2 y + 1)!}{(y + 1)! y!} = 10

\Rightarrow \frac{(y + y + 1)!}{(y + 1)! y!} = 10

\Rightarrow \frac{(y + 1 + 1) (y + 1 + 2) \dots (y + 1 + y)}{1 \times 2 \times \dots y} = 10

= (\frac{y + 1}{1} + 1) (\frac{y + 1}{2} + 1) \dots (\frac{y + 1}{y} + 1) = 10

s i n c e e a c h t e r m i s > 2, w e m a y h a v e

a t m o s t 3 t e r m s t o g e t p r o d u c t 10,

i . e . y ⩽ 3 .

w i t h y = 3 : \Rightarrow x = 8

\overset{x - 1}{} C_{y} =^{7} C_{3} = 35 \neq 10

w i t h y = 2 : \Rightarrow x = 6

\overset{x - 1}{} C_{y} =^{5} C_{2} = 10 \Rightarrow o k

\Rightarrow s o l u t i o n i s x = 6, y = 2

Commented by Tawa1 last updated on 30/Mar/19

God bless you sir . I appreciate .

Commented by peter frank last updated on 30/Mar/19

n i c e w o r k